Elsevier

Information Sciences

Volume 507, January 2020, Pages 298-312
Information Sciences

Logit choice models for interactive attributes

https://doi.org/10.1016/j.ins.2019.08.013Get rights and content

Highlights

  • New probabilistic discrete choice models.

  • Representation of the complex decision making attitude.

  • Consideration of interaction among the criteria.

  • Variants of the proposed logit models.

  • Real Application.

Abstract

In multi attribute decision making (MADM), the choice of a decision-maker (DM) is often the result of an interplay between the DM’s unique attitude, the utilities (specific to the DM) for the attributes, and the degree of positive (or negative) interaction among the attributes. Based on this conglomeration of information, we introduce a few choice models to give the probabilities for different possible choices of a DM. More specifically, the conventional multinomial logit (MNL) model is extended to consider the interaction among the attributes, and also the DM’s unique attitudinal character. The usefulness of the proposed models is illustrated in a real world application.

Introduction

Multinomial logit (MNL) [27] model is by far the most widely used discrete choice model. It postulates that each alternative can be seen as a bundle of attributes desired by a decision-maker (DM) who makes choices among various alternatives so as to maximize his/her utility by choosing the alternative whose attributes collectively yield more utility than those of all other alternatives.

Let us consider a situation in which DM chooses an alternative among multiple alternatives. Let ith alternative is denoted by Ai, which is characterized by a set of M attributes (c1,,cM). Let ai(m) denote mth attribute of Ai. The conventional choice models such as MNL considers the subjective utility corresponding to ai(m) as:vi(m)=β(m)ai(m),where β(m) is the utility coefficient that determines the DM’s unique evaluation scheme for cm. The net utility corresponding to the given attributes is termed as representative utility, and is computed as:Vi=m=1Mβ(m)ai(m)

MNL model considers that the choices of a DM cannot be fully explained by Vi, because of the presence of unobservable attributes. The total utility from Ai is considered to be given as:Ui=Vi+ϵiwhere ϵi is the utility from unobservable attributes. Since ϵi is unknown, one cannot determine a DM’s choice with certainty. Different choice models have appeared, considering different assumptions about the distribution of ϵ over the given alternatives.

MNL model considers ϵ as an i i d (independent and identically distributed) extreme random variable for all i, with a Gumbel distribution. Such an assumption is restrictive but it leads to very convenient form of choice probability with a closed-form, and is shown as:Pi=exp(Vi)i=1Kexp(Vi)=exp(m=1Mβ(m)ai(m))k=1Kexp(m=1Mβ(m)ak(m))In order to avoid i i d assumption, MNL model has been extended to other models such as nested logit [13], [42], cross-nested logit [39], probit [12], and mixed logit [28] models.

The popularity of the discrete choice models can be gauged through their applications in diverse domains in the recent times. They are applied in severity analysis [9], [25], [44], price optimization [41], revenue optimization [15], location planning [19], choice analysis problems [18], [22], [23], [30], [31], risk analysis [3], [5], [6], [7], [21], [40], demand analysis [14], [38], data analytics [4], [8], [17], regression analysis [11], [24], [32], causal inference in medicine [34], and forecasting [20], to name a few.

All these models are based on the premise of the utility maximizing behaviour [36]. The net utility that a decision-maker (DM) derives from an alternative depends on the evaluations of the utilities the DM derives from the multiple attributes of the alternative. The DM chooses the alternative that yields the maximum utility to him. Any decision or choice by a DM is an outcome of a complex process that involves:

  • evaluation of the multi-attribute utilities that are specific to a DM, and

  • aggregation of these utilities to arrive at the net utility corresponding to an alternative.

The DM chooses the alternative that yields the maximum utility to him as a result of the aggregation of the attributes evaluations.

In multi attribute decision making, the each alternative is evaluated against a set of attributes. In fact, we often evaluate an alternative against a collection of attributes, and not against each attribute individually, as assumed in the extant models. For instance, an employer evaluates a prospective employee taking his many credentials together. The candidate’s different attributes (skills/credentials), in reality, influences positively or negatively his other attributes.

The existing choice models of the likes of logit and probit, focus on the evaluation of the attribute utilities, but not on this aspect. More specifically, they model the evaluation of attribute utilities through the utility coefficients. The attribute utilities are then simply added to obtain the representative utility. However, in practice, the human aggregation is often not an additive operation, as demonstrated in several studies  [1], [35], [45], [46], [47].

In fact, there often exists an interaction (synergy or redundancy) among the attribute values. Besides, each individual displays a varying degree of tolerance (or compensation) in his aggregation process. Compensation refers to the degree by which a bad score on one attribute can be compensated by a good score on another attribute. For instance, AND and OR aggregation are the extreme cases giving the minimum and the maximum of the arguments of aggregation, with zero and full compensation respectively.

In the sequel, we refer to this varying compensation in a DM’s aggregation process as his unique attitudinal character. Both the degree and type (positive/negative) of attributes interaction, and the compensation degree, are specific to a DM, and hence bound to play an important role in the choice behaviour. Our objective in this work is to add to the excellent abilities of the choice models by focusing on these aspects of the human aggregation process.

In this process, our endeavor is also to bring the research advancements in the respective areas of aggregation operators and the choice modelling useful to each other. More specifically, the study of aggregation operators is a well established subfield of OR and decision sciences. Discrete choice modelling is an important topic in econometrics, used for predicting the choices of a DM using an empirical-quantitative approach. All such models need to aggregate the individual attribute values to arrive at the net utility of an alternative. The present work is based on exploiting this complementarity between these two research areas.

Broadly speaking, the present work adds to the capabilities of the extant choice modelling techniques through the recent advances in the area of aggregation operators. More specifically, we extend the extant probabilistic models of discrete choice to consider interaction among the attributes, and a DM’s attitudinal character at the same time. Unlike the extant models, the proposed models address the situations where different DMs, with the same attribute evaluations, may still have the different choices, due to their individualistic aggregation processes. The state-of-the-art models always predict the same choice probability in such situations.

The rest of the paper is organized as follows: Section 2 builds the background. Section 3 elaborates upon the characteristics of the proposed choice models. Section 4 introduces Choquet multinomial logit model. Section 5 further extends the proposed logit model with a consideration of the agent’s attitude. Section 6 is dedicated to a real world case-study, and Section 7 gives the conclusions.

Section snippets

Background

The recent aggregation operators offer a good potential in extending the capabilities of the existing choice models. In particular, we find Choquet integral and its variant attitudinal Choquet integral, very interesting. This section gives an overview of these operators that would be used to develop the proposed choice models.

Need for the proposed choice models

There are certain attributes of the human decision making that inevitably affect the final choices. This necessitates an explicit consideration of these attributes in choice modelling. This section elaborates upon these attributes that form the salient aspects of the proposed choice models. Broadly, these can be categorized as:-

  • i)

    the interaction among the attributes, and

  • ii)

    the decision-maker’s attitude.

Choquet multinomial logit model

We consider K alternatives A=(a1,,aK), each of which is defined by a set of M attributes. (c1,,cM). The set of values that ai takes is shown as:ai=(ai(1),,ai(M))

Each of these attribute values holds a different utility that is modelled by the vector:β=(β(1),,β(M))We shall first present Choquet multinomial logit model followed by its extension with a DM’s attitudinal character, and which is termed as attitudinal Choquet multinomial logit model.

We reconsider the utility model, as given in (2).

Attitudinal Choquet multinomial logit model

The proposed CMNL does not take into consideration the varying attitudes of the DMs. To this end, we extend the proposed CMNL model as attitudinal CMNL (ACMNL). Here, we compute Vi through the attitudinal Choquet integral as:Vi=logλ(m=1M(μ(B(m))μ(B(m+1)))λvi(σ(m))),and the ACMNL probability is shown as:Pi=exp(logλ(m=1M(μ(B(m))μ(B(m+1)))λvi(σ(m))))k=1Kexp(logλ(m=1M(μ(B(m))μ(B(m+1)))λvk(σ(m)))),where λ ∈ (0, ∞), λ ≠ 1 indicates a DM’s level of disjunctiveness. ACMNL model helps to cater to

Selection of the best car

We consider a real case-study about the selection of the best car among the latest car models [37]. Each of the models is described by a set of 4 attributes :- c1: length (mm), c2: width (mm), c3: height (mm), c4: engine capacity (cc). The original attribute values for the various car models are given in Table 2. The fuzzy measure, indicating the degree of interaction among the attributes for the buyer is shown in Table 1. The utility coefficients, corresponding to c1,,c4, specific to the

Conclusions

The proposed logit discrete choice models give the choice probabilities, based on the interaction among the attribute utility values and the DM’s individual attitudinal character. The adjustable parameters in the proposed models help to represent a fine range of attitudinal effects. The proposed models hold potential in several applications such as in studying the decision making behaviour of a large set of population, consumer behaviour, or to predict the response of a population to a new norm

CRediT authorship contribution statement

Manish Aggarwal: Conceptualization, Formal analysis, Writing - original draft.

Declaration of Competing Interest

The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript.

References (47)

  • U. Thole et al.

    On the suitability of minimum and product operators for the intersection of fuzzy sets.

    Fuzzy Sets Syst.

    (1979)
  • C. van Campen et al.

    Client demands and the allocation of home care in the netherlands. a multinomial logit model of client types, care needs and referrals

    Health Policy

    (2003)
  • R. Wang

    Capacitated assortment and price optimization under the multinomial logit model

    Oper. Res. Lett.

    (2012)
  • F. Ye et al.

    Comparing three commonly used crash severity models on sample size requirements: multinomial logit, ordered probit and mixed logit models

    Anal. Method Accid. Res.

    (2014)
  • H.J. Zimmermann et al.

    Latent connectives in human decision making

    Fuzzy Sets Syst.

    (1980)
  • H.J. Zimmermann et al.

    Decisions and evaluations by hierarchical aggregation of information

    Fuzzy Sets Syst.

    (1983)
  • M. Aggarwal

    Attitudinal choquet integrals and applications in decision making

    Int. J. Intell. Syst.

    (2018)
  • T. Astebro et al.

    More than a dummy: the probability of failure, survival and acquisition of firms in financial distress

    Eur. Manag. Rev.

    (2012)
  • Y. Bentz et al.

    Neural networks and the multinomial logit for brand choice modelling: a hybrid approach

    J. Forecast.

    (2000)
  • N.M. Boyson et al.

    Hedge fund contagion and liquidity shocks

    J. Finance

    (2010)
  • P. Changpetch et al.

    Selection of multinomial logit models via association rules analysis

    Wiley Interdiscip. Rev.

    (2013)
  • G. Choquet, Theory of Capacities, pp....
  • P. Congdon

    Multinomial and Ordinal Regression Models Bayesian Statistical Modelling

    (2007)
  • Cited by (13)

    • Discrete choice models with Atanassov-type intuitionistic fuzzy membership degrees

      2023, Information Sciences
      Citation Excerpt :

      In the MNL model, the influence of unknown factors in the calculating of utility is modelled by a stochastic variable, and the selecting probability of an alternative is positively related to its utility value. Because of the convenience in explaining and operating, the MNL model is very popular and has been applied to severity analysis [14,15], facility location [16], assortment optimization [17,18], accident analysis [19,20], etc. In the traditional probabilistic models of discrete choice, including MNL, the utilities of alternatives on different attributes are expressed by using precise values.

    • Stated choice analysis of preferences for COVID-19 vaccines using the Choquet integral

      2022, Journal of Choice Modelling
      Citation Excerpt :

      Yet, the Choquet integral has received limited attention in discrete choice analysis. Aggarwal (2020) incorporate the Choquet integral into a multinomial logit model. Similarly, Tehrani et al. (2012) formulate a logistic regression model based on the Choquet integral.

    • Analysis and prediction of charging behaviors for private battery electric vehicles with regular commuting: A case study in Beijing

      2022, Energy
      Citation Excerpt :

      To further predict the BEVs charging behaviors, one-month data points are randomly selected to feed into the NL model, of which approximately two-thirds are used for training, and the remaining data are test set. Furthermore, we employ the MNL model as a comparison method [45]. Note that candidate variables with p < 0.1 for identifying significant influencing factors are included in the prediction model.

    • A multinomial probit model with Choquet integral and attribute cut-offs

      2022, Transportation Research Part B: Methodological
      Citation Excerpt :

      We identify two main limitations of the literature on CI applications and address them in this study. First, many studies have explored the application of CI in RUM-based logit models (Aggarwal, 2018, 2019, 2020; Büyüközkan et al., 2018; Demirel et al., 2017), but fail to develop econometrically-sound estimators and restrict substitution effects. In contrast, accommodating unrestrictive substitution effects is straightforward in the MNP-CI model.

    • Selecting green third party logistics providers for a loss-averse fourth party logistics provider in a multiattribute reverse auction

      2021, Information Sciences
      Citation Excerpt :

      For example, although a higher-quality delivery service may promote customer satisfaction, it also makes the 4PL to pay more money. On the other hand, an interaction (synergy or redundancy) may have occurred across attributes [1]. If a synergy exists, then the net importance of attributes is higher than the sum of the individual importance of each attribute, whereas the net importance of attributes decreases if redundancy exists.

    View all citing articles on Scopus
    View full text